Non-unitarity of CKM matrix from the vector singlet quark mixing and neutron electric dipole moment

5 Pages·0.13 MB·English

by  Yi Liao

Non-unitarity of CKM matrix from vector singlet quark mixing and neutron electric
dipole moment
Yi Liao∗
Department of Physics, Tsinghua University, Beijing 100084, P.R.China
Xiaoyuan Li
Institute of Theoretical Physics, The Chinese Academy of Sciences, Beijing 100080, P.R.China
In the standard model (SM) the lowest order contribution to the quark electric dipole moment
(EDM)occursatthethreelooplevel. Weshowthatthenon-unitarityoftheCKMmatrixinmodels
with an extended quark sector typically gives rise to a quark EDM at thetwo loop level which has
no GIM-like suppression factors except the external quark mass. The induced neutron EDM is of
1 order 10−29 e cm and can be well within the reach of the next generation of experiments if it is
0 further enhanced by long distance physicsas happensin theSM.
0
Keywords: electric dipole moment, CKM matrix, singlet quark
2
PACS numbers: 11.30.Er 12.15.Ff 12.60.-i 13.40.Em

n
a
J

6 Animportanttargetofparticlephysicsisthe determi- looplevelbecausethe relevantamplitudesdonotchange

nation of the Cabibbo-Kobayashi-Maskawa (CKM) ma- thequarkflavorandeachCKMmatrixelementisaccom-
2
trix[1],whichparameterizesthechargedcurrentinterac- panied by its complex conjugate so that no T-violating
v
3 tions of quarks. In the standard model (SM) with three complex phase can arise. At the two loop level indi-
6 generations of quarks the CKM matrix is a 3×3 uni- vidual diagrams can have a complex phase, but it has
0 tarymatrix,andthe CP violationis due to the presence been shown by Shabalin [10] that their sum vanishes
5
of a nonzero phase in this matrix. The unitarity of the strictly. The null result was confirmed afterwards by
0
CKM matrix is essential in suppressing flavour changing several groups [11]. It is thus thought that in the SM
0
0 neutralcurrent(FCNC)processes[2][3]. UsingtheCKM the lowest order contribution to quark EDM’s occurs at
/ unitarityrelationtheCP violatingphaseinformationcan thethree looplevel. Arecentcalculation[12]showsthey
h
be elegantly displayedin terms of unitary triangles. The are of order 10−35 to 10−34 e cm for u and d quarks.
p
- most interesting challenge of the B-factories now enter- The extreme smallness of the quark and neutron EDM’s
p
ing into operation, and of future collider B experiments in the SM makes them particularly suited for searching
e
is to try to pin down the angles in the unitary triangles for new physics. The current experimental upper bound
h
: [4]. If the measured angles violate either of the ‘triangle [13] [14], |d(n)|<6.3×10−26e cm, has put very strigent
v
conditions’, or correspond to a point (ρ,η) [5] which is constraints on extensions of the SM, such as additional
i
X outsidetheallowedregion,thenwewillhaveevidencefor Higgs fields, right-handed currents, or supersymmetric
r new physics. partners [15]. We reanalyzed the problem in Refs. [16]
a
Violation of the CKM unitarity can appear in models and [17] and found that the complete two loop cancella-
withanextendedquarksector[6]. Variousconstraintson tion canbe attributed to two special features in the SM:
the possibility that exotic quarks mix with the ordinary the purely left-handed structure of the charged current
SM quarks have been derived from low energy charged and the unitarity of the 3×3 CKM matrix. The cancel-
and neutral current phenomenology, Z physics, FCNC lationintroducedby the unitarityis the flavourdiagonal
processes andCP violationin neutral mesonsystems [7] analog of the GIM suppression in the FCNC processes,
[8][9]. Inthis letter we proposeto examine the unitarity with the mere difference being that the cancellation is
of the CKM matrix in a different setting, namely, by in- complete in the EDM case. Thus it is expected that in
vestigatingpossibleunitarityviolatingeffectsontheneu- models with an extended quark sector the contributions
tron electric dipole moment (EDM). We shall find that to quark EDM’s at the two loop level are no longer can-
informationfromthe neutronEDMis complementaryto celledthoroughlybecauseofviolationoftheCKMunitar-
that from FCNC processes in B physics and serves as a ity, and that potentially large EDM’s for quarks and the
self-consistency check of the relevant theory. neutroncanthen be induced. Ignoringpossible logarith-
In the minimal SM quark EDM’s vanish at the one mic factors, they are of order, d(q) ∼ eg4π−4δ˜mq/m2W,
∗Present mailing address: Institut fu¨r Theoretische Physik, Universit¨at Leipzig, Augustusplatz 10/11, D-04109 Leipzig,
Germany
1
where g is the semi-weak coupling, δ˜is the rephasing in- is no W±G∓A coupling and the W+W−A coupling is
variant measure of CP violation [18]. Note that there also very simple [16].[ A is the background electromag-
are no GIM-like suppression factors except the external netic field and G∓ are would-be Goldstone fields. ] We
light quark mass which is required by chirality flip. Nu- useGreekandLatinletterstodenoteup-anddown-type
merically they are of order 10−29 e cm, well within the quarksrespectively,andtheexternalquarkisdenotedas
reach of the next generation of experiments [19] if they u ord . Therearefourgroupsofcontributions,denoted
e e
arefurtherenhancedbylongdistancephysicsashappens as WW, WG, GW and GG, where the first and second
in the SM. letters refer to the bosons exchanged in the outer and
Weconsiderhereamodelwithoneextrasingletdown- inner loops respectively.
type quark in a vector-like representation of the SM
gauge group, SU(3) ×SU(2) ×U(1) . In addition to
C L Y
the three quark generations,each consisting of the three
representations(i=1,2,3) e j (cid:11) k e
Qi (3,2) , ui (3,1) , di (3,1) , (1) ((cid:11)) (i) ((cid:12))
L +1/6 R +2/3 R −1/3
we have the following vector-like representation:
FIG. 1. The Feynman diagram that contributes to the
d4(3,1)−1/3+d¯4(¯3,1)+1/3 (2) EDM of the up-type quark ue. A background electromag-
netic field is understood to be attached to internal lines in
Such a quark representationappears, for example, in E6 all possible ways. The dashed lines represent W± and G±
GUTs [20]. The model can be considered as a minimal bosons. Thediagram for thedown-typequarkde is obtained
extension of the SM in the sense that there is no other bysubstitutions: α→i and j, k→α, β.
changeinthegaugeandscalarsectors. Inparticular,the
charged currents remain purely left-handed and the CP Consider for example the ue quark EDM. We found
violation is still encoded in the CKM matrix. as in the SM [16] that the contribution from WW is
After spontaneous symmetry breaking, the down-type automaticallycancelledwithoutusinganyunitaritycon-
singlet quark (d ) mixes with the ordinary three down- ditions while other contributions have the following sep-
4
type quarks so that the weak and mass eigenstates are arate structure:
related by a 4×4 unitary matrix,
V V∗ V V∗[H (k)−H (j)], (4)
ek αk αj ej α α
d′ V V V V d
ud us ub u4 where H (k) is a function of masses of u , d , u and
α α k e
 s′  = Vcd Vcs Vcb Vc4  s  . (3) W±. The crucialpointis thatthe dependence ondj and
b′ V V V V b
   td ts tb t4   dk masses splits up. This is mainly responsible for the
 d′4 L  V0d V0s V0b V04 d4 L thoroughorpartialcancellationoccurringintheSMand
beyond. Summing up the pair of mirror-reflected dia-
In the basis where the up-type quarks are diagonalized,
grams (i.e., j ↔ k) doubles the imaginary part of the
the submatrix consisting of the first three rows in the
productofCKMmatrixelementswhileremovingitsreal
abovematrixappearsinSU(2) W andZ couplings,and
L part. The summation over six pairs of (jk), since we
it is the generalized CKM matrix in this model. Note
havefourdown-typequarksd ,d ,d andd,completely
i j k l
that there still exist unitarity relations among the up-
cancelscontributionsamongstthemselvesduetotheuni-
type quarks although the matrix is no longer unitary.
tarityofthe4×4matrixV. Forexample,theH (j)term
α
The non-unitarity of the matrix leads to FCNC Z cou-
is
plings amongst down-type quarks which have important
repercussions on FCNC and CP violating processes [8] 2iIm[VejVα∗j(Ve∗kVαk+Ve∗lVαl+Ve∗iVαi)] (5)
[9]. In this work we shall focus on the purely charged = 2iIm[VejVα∗j(δeα−Ve∗jVαj)]=0.
current contributions to the EDM’s. We shall also dis-
Inother words,the up-type quarkEDM’s vanishstrictly
cussbrieflythemorecomplicatedcontributionsinvolving
inthemodelwithanextradown-typesingletquarkasin
FCNC interactions.
theSM.However,forthedown-typequarkEDM’sinthe
Let us consider the quark EDM [21]. The effec-
model, we have one less up-type quark in virtual loops
tive Lagrangian for the EDM interaction is defined as
tocompletethe unitaritycancellationsothatthecancel-
L =−id/2ψ¯γ σ ψFµν, where Fµν is the electromag-
eff 5 µν lation is not thorough. For example, summing over the
netic tensor and d is the EDM of the fermion ψ. Again,
pairs of (αβ) in the contribution to the d quark EDM
e
no T-violating complex phase can arise at the one loop
(see Fig. 1 for notations) and using the unitarity of V,
level. ThecontributingFeynmandiagramatthetwoloop
the H (α) term is
i
levelisshowninFig.1. Followingourpreviousworks[16]
[17], we use the background field [22]- [24] ( or the non- 2iIm[V∗ V (V∗V +V∗V )]
αe αi βi βe γi γe (6)
linear [25] ) R gauge with ξ = 1. In this gauge, there = 2iIm[V V∗V V∗],
ξ αe αi 0i 0e
2
whichisgenerallynon-vanishing. Hereuα,uβ anduγ are d(d)heavy = Im[VtdVt∗4V04V0∗d]· 10mMdeV
the three up-type quarks. Therefore, the d quark EDM ·(−5.3,−0.86,+2.3)×10−26 e cm,
e
is proportional to, for m =(200,300,400)GeV , (10)
d4
d(d) = Im[V V∗V V∗]· md
light td t4 04 0d 10 MeV
2i Im[VαeVα∗iV0iV0∗e]Hi(α). (7) ·3.3×10−25 e cm.
Xi,α
Note that the same combination of CKM elements is in-
volved in the two contributions. Since the ‘heavy’ part
Note that to avoid complete cancellation due to
(V∗V )=δ =0, H (α) must involve the d mass. is generally smaller by one order of magnitude we retain
i αi 0i α0 i i
belowonlythe‘light’partwhichisindependentofthed
PNow let us evaluate analytically the d quark EDM, 4
e
mass. The product of CKM elements can be expressed
d(d ). ThisisfacilitatedbythemasshierarchyintheSM,
e
in terms of the 3×3 submatrix elements, e.g.,
m ≫ m ≫ m , where q stands for other five quarks,
t W q
andthe assumptionthat md4 ≫mt. We shoulddiscrim- Im[VtdVt∗4V04V0∗d]=Im[VtdVt∗bZbd]−Im[VtsVt∗dZds], (11)
inate two kinds of down-type quarks with d heavy and
4
where Z = V V∗ +V V∗ +V V∗, i,j = d,s,b are
others light, as well as two kinds of up-type quarks with ij ui uj ci cj ti tj
top heavy and others light. We want to retain only the precisely the couplings appearing in the FCNC Z inter-
terms that are least suppressed by light quark masses. actions amongst ordinary down-type quarks. The most
stringent bounds on them come from the neutral meson
Since the charged current is purely left-handed, the chi-
rality flip needed for the EDM operator has to be made mixingandFCNCdecays. Hereweadopttheboundsob-
tained by requiring that the new tree level FCNC effects
by the external quark mass. We found that H (α) is
i
proportionalto m2 for all of WG, GW and GG contri- do not exceed the experimental values. Some bounds
uα
butionswhenu islight. Thereforeweonlyneedtokeep mayberelaxedifdestructiveinterferenceoccursbetween
α
them and other contributions, e.g., the box diagrams.
the top quark in up-type quarks. The leading terms in-
We take |Z | ≤ 3×10−4, |Z | ∼ |Z | ≤ 10−3. These
volving the heavy d4 quark come from the WG and GG ds sb bd
contributions: bounds are actually interrelated with the extraction of
other elements like V and V . We do not attempt here
td ts
d(d ) = em G2m2 (4π)−4Im[V V∗V V∗] aglobalanalysiswhichisbeyondthemaininterestofthe
e heavy de F W te t4 04 0e
+QhdQ(cid:16)u−(cid:16)592993−−3198µµtt−+2µ43tµltnlnµµ4tµµ4t+−1013l6nlµnt(cid:17)µit(cid:17), (8) pnTurhemesneen,ritIcmawl[oVertskdt,Vimtb∗4uaVtt0e4s:Vim0|∗dVp]tld≤y|a2∼d×o0p1.t002−t,h5e,|Vaftonsld|lo∼win|Vgcbv|a∼lue0s.0fo4r.
m
where µ = m2/m2 and µ = m2 /m2 . The leading |d(d)light|≤6.5×10−30 d e cm. (12)
t t W 4 d4 W 10 MeV
terms involving the light d quarks are independent of
i Using the SU(6) relation for the neutron, we obtain
their masses so that we may use V∗V = −V∗ V
i αi 0i α4 04
m
to sum up their contributions andPobtain, |d(n)|≤0.8×10−29 d e cm. (13)
10 MeV
d(de)light = emdeG2Fm2W(4π)−4Im[VteVt∗4V04V0∗e] A few comments are in order.
Q −8+ 16π2 −12lnµ +8ln2µ (1) The non-unitarity of the CKM matrix generally
(cid:20) u(cid:18) 3 t t(cid:19) (9) induces FCNC interactions of the Z and Higgs bosons
+Q −4− 8π2 +4lnµ −8ln2µ , withdown-type quarks. We foundthat mixedexchanges
d(cid:18) 3 t t(cid:19)(cid:21) of W± and Z (or Higgs) can contribute to the EDM.
Since the Higgs boson decouples for heavy enough mass,
which comes from the GW contribution. We should em- we consider here the FCNC Z couplings. There are two
phasizethattheaboveresultsareobtainedinthelimitof types of diagrams. The first one is similar to Fig. 1 and
largemasshierarchysothatthecancellationoftheEDM its leading term for the d(d) contains the same matrix
in the degeneracy limit of down-type quarks cannot be elements as in Eq.(10)and decreases the above resultby
explicit therein. about 1/3 for m = 300GeV, while the d(u) is severely
d4
Wenotethatthede quarkEDM,d(de),occursatorder suppressed. The second type has a different topology
g4(mde/m2W) and there is no further suppression due to and involves the complicated issue of the renormaliza-
GIM mechanism [2]. This is typical for models without tion of the non-unitary CKM matrix. The details of all
CKMunitarity. Sometermsarefurther enhancedbythe this will be reserved for a future publication. The point
heavy top mass, while the absence of the heaviest m2 we want to make here is that the purely chargedcurrent
d4
enhancement is consistent with generalarguments based result in Eq.(13) gives us the correct order of magnitude
on gauge invariance and naive dimensional analysis [26]. ofthe neutronEDM since there is no plausible reasonto
For numerical analysis, we use the following param- expect strong cancellation between the charged current
eters: G = 1.2 × 10−5 GeV−2, m = 80 GeV and and FCNC contributions which generally involve several
F W
m =175 GeV. Then we have for the d quark, different Jarlskog rephasing invariants of CP violation.
t
3
(2) The above discussion can be easily generalized to [4] For a recent review, see for example, R. D. Peccei, hep-
other models with exotic quarks. At the two loop level ph/0004152.
both up- and down-type quark EDM’s vanish strictly in [5] L. Wolfenstein, Phys. Rev.Lett. 51(1983)1945.
the model with a sequential fourth generation. In the [6] For a recent review, see for example, Y. Nir, hep-ph
/9911321.
model with an extra up-type singlet quark, the down-
[7] P. Langacker and D.London, Phys.Rev. D38(1988)886;
type quark EDM’s vanish identically at two loop order.
E. Nardi, E. Roulet, and D. Tommasini, Phys. Lett.
Since all down-type quarks are light in this case, the
B344(1995)225; V. Barger, M.S. Berger and R.J.N.
leading terms for the u EDM must be proportional to
e Phillips, Phys. Rev. D52(1995)1663; F. del Aguila, J.
m m2 and are thus very small compared to the case
ue di A. Aguilar-Saavedra and R. Miquel, Phys. Rev. Lett.
considered above. 82(1999)1628; M. Popovic and E. Simmons, Phys. Rev.
(3)We havealsostudied the P andT violating purely D62(2000)035002.
gluonic operators, e.g., the dimension-6 Weinberg oper- [8] Y. Nir and D. Silverman, Nucl. Phys. B345(1990)301;
ator [27]. We found that they are severely suppressed Phys. Rev. D42(1990)1477; D. Silverman, Phys. Rev.
D45(1992)1800; G.C. Branco et al., Phys. Rev.
by light quark masses in the current case as in the
D48(1993)1167; W.-S. Choong and D. Silverman, Phys.
SM [28]. Their contribution to the neutron EDM can
Rev. D49(1994)2322; L. Lavoura and J.P. Silva, Phys.
be ignored. To obtain the quark chromoelectric dipole
Rev. D47(1993)1117; D.Silverman, Int. J. Mod. Phys.
moment(CEDM), one merely replaces eQ by g in
u,d s A11(1996)2253; M. Gronau and D. London, Phys. Rev.
Eqns.(8) and (9) with g being the QCD coupling. It
s D55(1997)2845; G. Barenboim and F.J. Botella, Phys.
is thenclearthatthe quarkCEDMis muchsmallerthan Lett. B433(1998)385; G. Barenboim, F.J. Botella, G.C.
thequarkEDMsothatthelatterremainstobethedom- BrancoandO.Vives,Phys.Lett.B422(1998)277; D.Sil-
inant contribution in the neutron EDM. verman, Phys. Rev. D58(1998)095006; G. Eyal and Y
Nir,JHEP9909(1999)013;K.HiguchiandK.Yamamoto,
(4)In the static limit the fermionic part ofthe EDM is
Phys. Rev.D62(2000)073005.
identicaltothespinoperator. Sinceasignificantamount
[9] G. Bhattacharyya, G. Branco and D. Choudhury, Phys.
of the proton spin is derived from the polarized strange
Lett. B336(1994)487; B340(1994)266(E); L. T. Handoko
quark sea [29], it seems reasonable that the strange
andT.Yoshikawa,Int.J.Mod.Phys.A12(1997)5609;C.-
quarkalsocontributesto the neutronEDM[30]. Forthe
H. V. Chang, D. Chang and W.-Y. Keung, Phys. Rev.
strange quark EDM, we have enhancement factors from D61(2000)053007.
masses and CKM elements. For the latter the dominant [10] E. P.Shabalin, Sov. J. Nucl. Phys.28(1978)75.
one is Im[VtbVt∗sZsb]. Suppose |η| ∼ 10% of the proton [11] I. B. Khriplovich, Phys. Lett. B173(1986)193; Sov. J.
spin is accounted for by the strange sea, then we have Nucl. Phys. 44(1986)659; A. Czarnecki and B. Krause,
|d(n) |≤0.8×10−29(m /m )(|V |/|V |)|η|e cm∼ Acta Phys. Pol. B28(1997)829.
strange s d ts td
3.2×10−29e cmforthequotedparametersandm =200 [12] A. Czarnecki and B. Krause, Phys. Rev. Lett.
s
78(1997)4339.
MeV.
[13] K. F. Smith et al, Phys. Lett. B234(1990)191; I. S.
(5) Another possible enhancement for the neutron
Altarev et al, Phys.Lett. B276(1992)242.
EDM originates from long distance physics [31]. Al- [14] P. G. Harris et al, Phys.Rev.Lett. 82(1999)904.
though there are still controversies concerning this, it [15] For reviews, see, S. M. Barr and W. J. Marciano, in:
seems reliable to get an enhancement factor of two or- C.Jarlskog(Ed.),CPViolation,WorldScientific,Singa-
ders of magnitude in the SM. If this persists in the case pore,1989;andX.-G.He,B.H.J.McKellar,S.Pakvasa,
consideredhere,improvementofthe upperboundonthe Int.J. Mod. Phys. A4(1989)5011.
neutronEDMinthenearfuture willprovideaninterest- [16] Y. Liao and X. Li, Phys.Rev.D60(1999)073004.
ing test of the unitarity of the CKM matrix which will [17] Y. Liao and X. Li, Phys.Rev.D61(2000)076002.
[18] C.Jarlskog,Phys.Rev.Lett.55(1985)1039;O.W.Green-
becomplementarytoorevencompetitivewiththebound
berg, Phys. Rev. D32(1985)1841; D.-D. Wu, Phys. Rev.
from B physics.
D33(1986)860.
The workofX. Liwassupportedinpartby the China [19] R. Golub and K. Lamoreaux, Phys. Rep.237(1994)1.
National Natural Science Foundation under grant Num- [20] J. L. Rosner, hep-ph/9907438 and references cited
bers 19835060and 19875072. therein.
[21] The quark EDM in the model with a singlet quark has
been discussed before where CP violation is due to the
complex phase in the Higgs VEV’s. See, F. del Aguila
and J. Cortes, Phys. Lett. 156B(1985)243; L. Bento and
G.Branco,Phys.Lett.B245(1990)599; I.KakebeandK.
Yamamoto, Phys. Lett.B416(1998)184.
[1] N.Cabibbo,Phys.Rev.Lett.10(1963)531;M.Kobayashi [22] L. F. Abbott, Nucl. Phys. B185(1981)189; Acta Phys.
and T. Maskawa, Prog. Theo. Phys. 49(1973)652. Pol. B13(1982)33, and references cited therein.
[2] S. L. Glasow, J. Illiopoulos and L. Maiani, Phys. Rev. [23] M.B.EinhornandJ.Wudka,Phys.Rev.D39(1989)2758;
D2(1970)1285. A. Denner, G. Weiglein and S. Dittmaier, Phys. Lett.
[3] S.L. Glashow, S.Weinberg, Phys. Rev.D15(1977)1958. B333(1994)420.
4
[24] X.LiandY.Liao,Phys.Lett.B356(1995)68; A.Denner, [30] A. Zhitnitskii and I. Khriplovich, Sov. J. Nucl. Phys.
G.WeigleinandS.Dittmaier,Nucl.Phys.B440(1995)95. 34(1981)95; V. Khatsymovsky, I. Khriplovich and A.
[25] K.Fujikawa, Phys.Rev.D7(1973)393. Zhitnitskii, Z. Phys. C36(1987)455; X.-G. He, B. McK-
[26] Y. Liao, Y.-P. Kuang and C.-S. Li, Phys. Rev. eller ans S. Pakvasa, Phys. Lett. B254(1991)231; I.
D54(1996)3503. Khriplovich, Phys. Lett. B382(1996)145 C. Hamzaoui
[27] S.Weinberg, Phys. Rev.Lett. 63(1989)2333. and M. Pospelov, Phys.Rev. D60(1999)036003.
[28] M. Booth, Phys. Rev. D48(1993)1248; M. Pospelov, [31] M. B. Gavela et al, Phys. Lett. B109(1982)215; I.
Phys.Lett. B328(1994)441. B. Khriplovich and A. R. Zhitnitsky, Phys. Lett.
[29] D.KaplanandA.Manohar,Nucl.Phys.B310(1988)527; B109(1982)490; I. Bigi and N. G. Uraltsev, Sov. Phys.
X.Li and Y. Liao, Phys. Lett.B379(1996)219. JETP 73(1991)198.
5